An inverse subsemigroup T of a regular semigroup S is said to be an inverse transversal of S if T contains precisely one inverse of each element of S. Constructions based on the regular semigroup consisting of all regular elements of a Rees matrix semigroup over an inverse semigroup are used to establish the paper's main results. It is shown that a regular semigroup S contains an inverse transversal T which is a quasi- ideal of S (i.e., TS\(T\subseteq T)\) if and only if S can be embedded as an ideal in a locally inverse semigroup L which contains an idempotent u for which fu \({\mathcal R} f {\mathcal L} uf\) for each \(f=f^ 2\in S\). A construction is given which produces a regular semigroup with a given quasi-ideal inverse transversal T from T and two bands, one left normal, the other right normal, which contain a common quasi-ideal inverse transversal isomorphic to the semilattice of idempotents of T. Any regular semigroup with a quasi-ideal inverse transversal can be so constructed.